y''+2y' = 2x+5-e^(-2x)

y'' + 2y' = 2x + 5 - e^-2x

Step1:

Solve y'' + 2y' = 0

characteristic equation: r² + 2r = 0

r(r+2) = 0

r = 0 or r = -2

General solution:

y_h = c1 + c2 e^-2x

Step2: Find particular solution

1) 2x + 5:

y_p1 = Ax² + Bx

y' = 2Ax + B

y'' = 2A

y'' + 2y = 2A + 2(2Ax+ B) = 4Ax + 2B + 2A

4Ax + 2B + 2A = 2x + 5

4A = 2

A = 1/2

2A + 2B = 5

1 + 2B = 5

B = 2

y_p1 = x²/2 + 2x

2) -e^-2x:

y_p2 = Ae^-2x

y' = -2Ae^-2x

y'' = 4Ae^-2x

y'' + 2y' = 4Ae^-2x - 4Ae^-2x = 0

y_p2 = Axe^-2x

y' = Ae^-2x - 2Ax^-2x

y'' = -2Ae^-2x - 2Ae^-2x + 4Axe^-2x = (4Ax - 4A)e^-2x

y'' + 2y' = (4Ax - 4A)e^-2x + 2(Ae^-2x - 2Axe^-2x) = (-2A)e^-2x

-2Ae^-2x =  -e^-2x

A = 1/2

y_p2 = xe^-2x/2

Step3: General solution

y(x) = y_h + y_p1 + y_p2 = c1 + c2 e^-2x + x²/2 + 2x + xe^-2x/2

y(x) = y_h + y_p1+ y_p2 = c1 + c2 e^-2x + x²/2 + 2x + xe^-2x/2